Olympiad problems

2018 Junior Balkan Team Selection Tests - Romania, 2

Let $k > 2$ be a real number. a) Prove that for all positive real numbers $x,y$ and $z$ the following inequality holds: $$\sqrt{x + y }+\sqrt{y + z }+\sqrt{z + x} > 2\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$ b) Prove that there exist positive real numbers $x, y$ and $z$ such that $$\sqrt{x + y }+\sqrt{y + z}+\sqrt{z + x} <k\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$ Leonard Giugiuc

2019 Balkan MO Shortlist, A2

Tags: functional equation , algebra , AZE IMO TST , BMO Shortlist , TST

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(xy) = yf(x) + x + f(f(y) - f(x)) \] for all $x,y \in \mathbb{R}$.

2024 Azerbaijan IZhO TST, 4

Tags: number theory , TST , AZE IzHO TST

Take a sequence $(a_n)_{n=1}^\infty$ such that $a_1=3$ $a_n=a_1a_2a_3...a_{n-1}-1$ [b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence. [b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.

2023 Chile TST Ibero., 2

Tags: TST , Chile , algebra

Consider a function $ n \mapsto f(n) $ that satisfies the following conditions: $ f(n) $ is an integer for each $ n $. $ f(0) = 1 $. $ f(n+1) > f(n) + f(n-1) + \cdots + f(0) $ for each $ n = 0, 1, 2, \dots $. Determine the smallest possible value of $ f(2023) $.

2023 Chile TST Ibero., 3

Tags: TST , Chile , number theory

Determine the smallest positive integer $ n $ with the following property: for every triple of positive integers $ x, y, z $, with $ x $ dividing $ y^3 $, $ y $ dividing $ z^3 $, and $ z $ dividing $ x^3 $, it also holds that $ (xyz) $ divides $ (x + y + z)^n $.

2018 Azerbaijan IZhO TST, 1

Tags: algebra , TST , AZE IzHO TST

Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

2025 Taiwan TST Round 1, N

Tags: number theory , TST

Let $k$ be a positive integer. A positive integer $n$ is called a $k$-good number if it satisfies the following two conditions: 1. $n$ has exactly $2k$ digits in decimal representation (it cannot have leading zeros). 2. If the first $k$ digits and the last $k$ digits of $n$ are considered as two separate $k$-digit numbers (which may have leading zeros), the square of their sum is equal to $n$. For example, $2025$ is a $2$-good number because \[(20 + 25)^2 = 2025.\] Find all $3$-good numbers.

2017 China Team Selection Test, 1

Tags: TST , combinatorics , equation , China , generating functions

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

Azerbaijan Al-Khwarizmi IJMO TST 2025, 4

Tags: combinatorics , AZE Al-Khwarizmi IJMO TST , TST

The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b+1$ is written instead.What will be the number remained on the board after the last step.

2018 Azerbaijan JBMO TST, 2

Tags: algebra , Inequality , TST , AZE JBMO TST

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

2023 Hong Kong Team Selection Test, Problem 1

Tags: Inequality , TST , algebra , easy

Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

2023 Serbia JBMO TST, 1

Tags: geometry , TST

Given is an isosceles triangle $ABC$ with $CA=CB$ and angle bisector $BD$, $D \in AC$. The line through the center $O$ of $(ABC)$, perpendicular to $BD$, meets $BC$ at $E$. The line through $E$, parallel to $BD$, meets $AC$ at $F$. Prove that $CE=DF$.

2023 Chile TST IMO, 4

Tags: combinatorics , Chile , TST

On a $ 10 \times 10 $ chessboard, there are 91 white pawns placed in different squares. Nico picks a white pawn, paints it black, and places it in an empty square, repeating the process until all pawns have been painted. Prove that at some point, there will be two pawns of different colors placed on squares that share a common edge.

2017 VJIMC, 4

Tags: number theory , TST , AZE IzHO TST

A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.

2024 Azerbaijan BMO TST, 4

Tags: combinatorics , AZE BMO TST , TST

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

2018 Azerbaijan JBMO TST, 3

Tags: number theory , TST , AZE JBMO TST

Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

2023 Chile TST IMO, 1

Tags: TST , Chile , geometry

Let $ \triangle ABC $ be an equilateral triangle, and let $ M $ be the midpoint of $ BC $. Let $ C_1 $ be the circumcircle of triangle $ \triangle ABC $ and $ C_2 $ the circumcircle of triangle $ \triangle ABM $. Determine the ratio between the areas of the circles $ C_1 $ and $ C_2 $.

2016 Azerbaijan JBMO TST, 1

Tags: inequalities , algebra , TST , AZE JBMO TST

If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

2023 Balkan MO Shortlist, G5

Tags: geometry , AZE BMO TST , TST , Miquel point

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

2023 ISL, N2

Tags: number theory , IMO Shortlist , AZE IMO TST , TST

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

2003 USA Team Selection Test, 2

Tags: geometry , ratio , TST , USA

Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that \[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \] if and only if $P$ lies on at least one of the medians of triangle $ABC$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)

2024 Israel TST, P3

Tags: combinatorics , TST

Let $0<c<1$ and $n$ a positive integer. Alice and Bob are playing a game. Bob writes $n$ integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most $cn$ moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of $c$ does Alice win for all large enough $n$?

1995 China Team Selection Test, 2

Tags: geometry , geometric transformation , geometry unsolved , Tangents , Locus problems , China , TST

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

2022 Bolivia IMO TST, P1

Tags: Bolivia , algebra , TST , Gauss identity

Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy $$x^3+y^3+3x^2y^2=x^3y^3$$

2019 Iran RMM TST, 3

Tags: combinatorics , Tiling , TST , Iranian TST

An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other. Clarifications for network: It means an infinite board consisting of square cells.